Unipotent Representatons of E6(quat) in block dual to E6(F4)

G= E6(quaternionic) = compact inner class, quasisplit (adjoint)
G^vee= E6(F4) = split inner class, not split (simply connected)

      cpt  herm  quat
F4    0    0     45
split 1    513   1881

quaternionic	     E6(F4)
special		     special                     
orbit    cells	     dualorbit cells	diagram	 #O_R
E6	 0	     0	       2	000000	 1
E6(a1)	 1	     2A1       1	100001   (not even)
D5	 2	     2A2       0	200002	 1

#O_R is the number of real orbits of real points of (simply connected)
E6(F4). This group is connected, so this can be read off from the
tables in Collingwood-McGovern.

	         b
	         |
abcdef means a-c-d-e-f

Note: adjoint/simply connected don't matter for E6 since Z=Z/3.

Note: E6(F4) has only one Cartan (fundamental, (C^x)^2 x (S^1)^2, connected)
=> 

** Stability is empty for this block of E6(quaterninic) **

E6(F4)
cell    aq
0:      0,1,2,3,7,9
1:      22
2:      44	 

Summary: each cell for E6(F4) contains a distinguished A(lambda) (7,22,44)
Orbit 2A1 on dual side isn't even

Each (even) unipotent packet for E6(quaternionic) in the block is a
singleton, dual to the distinguished A(lambda)

E6(F4)	     E6(F4)           E6(quat)  E6(quat)
dual orbit   dual A(lambda)   cell      unipotent
0            44		      0		0 (large fundamental series at 0)
2A2	     7		      2		37

====================================================================
quaternionic	     E6(F4)
special		     special                     
orbit    cells	     dualorbit cells	diagram	 #O_R
E6	 0	     0	       2	000000	 1

cells	 stable sums
0	 0

%stable -d -S 1,2,3,4,5,6 -c 2

lambda is singular at simple roots: 1,2,3,4,5,6
cells:2
Parameters (living at lambda): 0
 0( 851,44):   8  4  [C+,rn,rn,rn,rn,C+]   2   0   0   0   0   1   ( *, *)  ( *, *)  ( *, *)  ( *, *)  ( *, *)  ( *, *)   2,3,4,2,3,4,5,4,2,3,4,5

Dual parameters (to those living at lambda): 44
44(44, 851):  12  0  [C-,ic,ic,ic,ic,C-]  42  44  44  44  44  43   ( *, *)  ( *, *)  ( *, *)  ( *, *)  ( *, *)  ( *, *)   1,3,4,2,5,4,3,1,6,5,4,2,3,1,4,3,5,4,2,6,5,4,3,1

Dimension of space of stable characters: 1
Everything is stable
-------------------------------------------------------------------
quaternionic	     E6(F4)
special		     special                     
orbit    cells	     dualorbit cells	diagram	 #O_R
D5	 2	     2A2       0	200002	 1

%stable -d -c 0 -S 2,3,4,5

lambda is singular at simple roots: 2,3,4,5
Parameters (living at lambda): 37
37(1762, 7):  17  4  [C-,C+,rn,rn,rn,C-]  30  41  37  37  37  32   ( *, *)  ( *, *)  ( *, *)  ( *, *)  ( *, *)  ( *, *)   1,3,1,4,2,3,1,5,4,2,3,1,4,3,6,5,4,2,3,1,4,3,5,4,2,6,5,4,3,1

Dual parameters (to those living at lambda): 7
 7( 7,1762):   3  0  [C+,C-,ic,ic,ic,C+]  14   3   7   7   7  12   ( *, *)  ( *, *)  ( *, *)  ( *, *)  ( *, *)  ( *, *)   2,4,3,5,4,2

Dimension of space of stable characters: 1
Everything is stable